How to make mesh lights brighter without washing out in the render?
You can do this by using the Is Camera Ray output of a Light Path node to dim the emission strength (or even use a different shader) for the camera.  The Is Camera Ray will output a 1 if the ray is coming from the camera (see here for an explanation of ray tracing) and a 0 otherwise.

You can plug this into the mix factor of a Mix Shader or Mix RGB node and plug what you want the camera to see in the bottom socket.



Here is an example, the left uses just the emission shader on the lamp, the right uses the above node trick.